C-SAMPLE TESTS OF HOMOGENEITY AGAINST ORDERED ALTERNATIVES

Abstract

Let population πi. have an unknown continuous cumulative distribution function Fi(x), (i = 1, 2, …, c). The problem considered is to test the null hypothesis, H0, that the Fi';s are equal against the alternative: Fl(x) ≥ F2(x) ≥ … ≥ Fc(x) (or F1(x) ≤ F2(x) ≤ … ≤ Fc(x)) For all x with at least one inequality strict. Nonparametric tests for this problem were proposed by Jonckheere (Biometrika41 (1945) 135–145), Chacko (Ann. Math. Statist. 34 (1963) 945–956), and Puri (Comm. Pure Appl. Math. 18 (1965) 51–63). These tests are for location shift alternatives and are based on the asymptotic distribution of the statistics. For tests of H0 against two classes of ordered alternatives of the Lehmann type, we obtain a locally most powerful rank order test and the test statistic is found to be a weighted sum of c-sample rank order statistics of the type introduced by Savage (Ann. Math. Statist.27 (1956) 590–615). We obtain a general result pertaining to the optimal weights to be assigned in a weighted sum of c-sample linear rank statistics such that the Pitman efficacy of the test procedure is maximized. Applying this result we propose a class of test statistics for location shift and change of scale alternatives which are weighted sums of rank order statistics of the type introduced by Chernoff-Savage (Ann. Math. Statist.29 (1958) 972–994), Bhapkar (Ann. Math. Statist.32 (1961) 1108–1117), and Deshpande (J. Indian Statist. Assoc.3 (1965) 20–29). These Chernoff-Savage statistics are easier to compute than and are asymptotically equivalent to those proposed by Puri (1965) for equally spaced location shift alternatives.